Additional Requirements

Aims

To give an introduction to Modern Analysis, including elements of functional analysis. The emphasis will be on ideas and results.

Overview

This course unit deals with a coherent and elegant collection of results in analysis. The aim of this course is to provide an introduction to the theory of infinite dimensional linear spaces, which is not only an important tool, but is also a central topic in modern mathematics.

This area has many applications to other areas in Pure and Applied Mathematics such as Dynamical Systems, C* algebras, Quantum Physics, Numerical Analysis, etc.

Learning outcomes

Learning Outcomes

On successful completion of this module students will be able to

understand the approximation of continuous functions using the Stone-Weierstrass Theorem;

understand the concepts of Hilbert and Banach spaces, with l 2 and l p spaces serving as examples;

define linear operators, their spectrum (with matrices serving as examples) and their spectral radius, understand the definitions of self-adjoint, isometric and unitary operators on Hilbert spaces and their spectra, be able to apply these ideas to matrices.

Students will have seen proofs of the main results, although an intimate knowledge of the full details is not required.

Future topics requiring this course unit

This course would be useful to students interested in the following topics: MATH41012 Fourier Analysis and Lebesgue Integration.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.